Polarization, sign sequences and isotropic vector systems

Gergely Ambrus and Sloan Nietert

Vol. 303 (2019), No. 2, 385–399

DOI: 10.2140/pjm.2019.303.385

Abstract

We determine the order of magnitude of the $n$-th ${\ell }_p$-polarization constant of the unit sphere $S^{d-1}$ for every $n,d\ge 1$ and $p>0$. For $p=2$, we prove that extremizers are isotropic vector sets, whereas for $p=1$, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for $d=2$, we discuss the optimality of equally spaced configurations on the unit circle.

Keywords

polarization problems, discrete potentials, Chebyshev constants, isotropic vectors sets, tight frames, vector sums

Mathematical Subject Classification 2010

Primary: 52A40

Secondary: 41A17

Milestones

Received: 23 April 2019

Revised: 19 June 2019

Accepted: 19 June 2019

Published: 4 January 2020